Question: Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle AOB = 4x + 20$, and $ m \angle BOC = 6x + 0$, find $m\angle AOB$. $O$ $A$ $C$ $B$
Answer: From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {4x + 20} + {6x + 0} = {90}$ Combine like terms: $ 10x + 20 = 90$ Subtract $20$ from both sides: $ 10x = 70$ Divide both sides by $10$ to find $x$ $ x = 7$ Substitute $7$ for $x$ in the expression that was given for $m\angle AOB$ $ m\angle AOB = 4({7}) + 20$ Simplify: $ {m\angle AOB = 28 + 20}$ So ${m\angle AOB = 48}$.